/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem 1: 

(VAR x y)
(RULES
double(0) -> 0
double(s(x)) -> s(s(double(x)))
minus(s(x),s(y)) -> minus(x,y)
minus(x,0) -> x
plus(0,y) -> y
plus(s(x),y) -> plus(x,s(y))
plus(s(x),y) -> s(plus(minus(x,y),double(y)))
plus(s(x),y) -> s(plus(x,y))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 DOUBLE(s(x)) -> DOUBLE(x)
 MINUS(s(x),s(y)) -> MINUS(x,y)
 PLUS(s(x),y) -> DOUBLE(y)
 PLUS(s(x),y) -> MINUS(x,y)
 PLUS(s(x),y) -> PLUS(minus(x,y),double(y))
 PLUS(s(x),y) -> PLUS(x,s(y))
 PLUS(s(x),y) -> PLUS(x,y)
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))

Problem 1: 

SCC Processor:
-> Pairs:
 DOUBLE(s(x)) -> DOUBLE(x)
 MINUS(s(x),s(y)) -> MINUS(x,y)
 PLUS(s(x),y) -> DOUBLE(y)
 PLUS(s(x),y) -> MINUS(x,y)
 PLUS(s(x),y) -> PLUS(minus(x,y),double(y))
 PLUS(s(x),y) -> PLUS(x,s(y))
 PLUS(s(x),y) -> PLUS(x,y)
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
->->-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->->Cycle:
->->-> Pairs:
 DOUBLE(s(x)) -> DOUBLE(x)
->->-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->->Cycle:
->->-> Pairs:
 PLUS(s(x),y) -> PLUS(minus(x,y),double(y))
 PLUS(s(x),y) -> PLUS(x,s(y))
 PLUS(s(x),y) -> PLUS(x,y)
->->-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 DOUBLE(s(x)) -> DOUBLE(x)
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Projection:
 pi(DOUBLE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 PLUS(s(x),y) -> PLUS(minus(x,y),double(y))
 PLUS(s(x),y) -> PLUS(x,s(y))
 PLUS(s(x),y) -> PLUS(x,y)
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
-> Usable rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[double](X) = 2.X + 1
[minus](X1,X2) = X1
[0] = 1
[s](X) = X + 1
[PLUS](X1,X2) = 2.X1

Problem 1.3: 

SCC Processor:
-> Pairs:
 PLUS(s(x),y) -> PLUS(x,s(y))
 PLUS(s(x),y) -> PLUS(x,y)
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x),y) -> PLUS(x,s(y))
 PLUS(s(x),y) -> PLUS(x,y)
->->-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))

Problem 1.3: 

Subterm Processor:
-> Pairs:
 PLUS(s(x),y) -> PLUS(x,s(y))
 PLUS(s(x),y) -> PLUS(x,y)
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Projection:
 pi(PLUS) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(0) -> 0
 double(s(x)) -> s(s(double(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(0,y) -> y
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(x),y) -> s(plus(x,y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.